Photovoltaic cells produce a voltage that varies with current, cell operating condition, cell physics, cell defects, and cell illumination. One mathematical model for a photovoltaic cell, as illustrated in FIG. 1, models output current as:
                    I        =                              I            L                    -                                    I              0                        ⁢                          {                                                exp                  ⁡                                      [                                                                  q                        ⁡                                                  (                                                      V                            +                                                          IR                              S                                                                                )                                                                    nkT                                        ]                                                  -                1                            }                                -                                    V              +                              IR                S                                                    R              SH                                                          EQN        .                                  ⁢        1            Where
IL=photogenerated current
RS=series resistance
RSH=shunt resistance
I0=reverse saturation current
n=diode ideality factor (1 for an ideal diode)
q=elementary charge
k=Boltzmann's constant
T=absolute temperature
I=output current at cell terminals
V=voltage at cell terminals
For silicon at 25° C., kT/q=0.0259 Volts.
Typical cell output voltages are low and depend on the band gap of the material used to manufacture the cell. Cell output voltages may be merely half a volt for silicon cells, far below the voltage needed to charge batteries or drive most other loads. Because of these low voltages, cells are typically connected together in series to form a module, or an array, having an output voltage much higher than that produced by a single cell.
Real-world photovoltaic cells often have one or more microscopic defects. These cell defects may cause mismatches of series resistance RS, shunt resistance RSH, and photogenerated current IL from cell to cell in a module. Further, cell illumination may vary from cell to cell in a system of photovoltaic cells, and may vary even from cell to cell in a module, for reasons including shadows cast by trees, bird droppings shadowing portions of a cell or module, dust, dirt, and other effects. These mismatches in illumination may vary from day to day and with time of day—a shadow may shift across a module during a day, and rain may wash away dust or dirt shadowing a cell.
From EQN. 1, output voltage is greatest at zero output current, and output voltage V falls off nonlinearly with increasing output current I. FIG. 2 illustrates the effect of increasing current drawn from a photovoltaic device at constant illumination. As current I is increased under constant illumination, voltage V falls off slowly, but as current I is increased to an output current near the photocurrent IL, output voltage V falls off sharply. Similarly, cell power, the product of current and voltage, increases as current I increases, until falling voltage V overcomes the effect of increasing current, whereupon further increases in current I drawn from the cell cause power P to decrease rapidly. For a given illumination, each cell, module, and array of cells and modules therefore has a maximum power point (MPP) representing the voltage and current combination at which output power from the device is maximized. The MPP of a cell, module, or array will change as temperature and illumination, and hence photo-generated current IL, changes. The MPP of a cell, module, or array may also be affected by factors such as shadowing and/or aging of the cell, module, or array.
Maximum Power Point Tracking (MPPT) controllers for operating a photovoltaic cell at or near its maximum power point have been proposed. These controllers typically determine an MPP voltage and current for a photovoltaic device connected to their input, and adjust their effective impedance to maintain the photovoltaic device at the MPP. However, conventional MPPT controllers often have one or more drawbacks. For example, some proposed MPPT controllers may be relatively slow under certain conditions, thereby delaying MPP operation.